Bifurcation Diagrams

Abstract

In the fifties, Myrberg (1958, 1959, 1963) discovered infinite cascades of period doubling bifurcations. The word “bifurcation” means a sudden qualitative change in the nature of a solution, as a parameter is varied. The parameter value at which a bifurcation occurs, is called a bifurcation parameter value. He found that as a parameter was varied, a fixed point attractor could bifurcate into an attracting period 2 orbit, which could again double to an attracting period 4 orbit, followed rapidly by an infinite sequence, period 8, 16, etc. Period doubling bifurcation will be described more carefully later in this section. Unfortunately, he did not have a computer to produce the pictures of these phenomena. In the seventies the first computer generated bifurcation diagrams appeared in the literature. The concept of bifurcation diagram includes a number of ways of plotting a phase variable on one axis and a parameter on another. The pages that follow explain the procedures for making a variety of bifurcation diagrams. What you want to display determines the kind of bifurcation diagram you require. As we will see in a moment, with the program it is rather easy to produce detailed bifurcation diagrams. These bifurcation diagrams show many sudden qualitative changes, that is, many bifurcations, in the chaotic attractor as well as in the periodic orbits.